Best Known (59, 86, s)-Nets in Base 5
(59, 86, 252)-Net over F5 — Constructive and digital
Digital (59, 86, 252)-net over F5, using
- 12 times m-reduction [i] based on digital (59, 98, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 49, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 49, 126)-net over F25, using
(59, 86, 589)-Net over F5 — Digital
Digital (59, 86, 589)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 589, F5, 27) (dual of [589, 503, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 624, F5, 27) (dual of [624, 538, 28]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(586, 624, F5, 27) (dual of [624, 538, 28]-code), using
(59, 86, 52657)-Net in Base 5 — Upper bound on s
There is no (59, 86, 52658)-net in base 5, because
- 1 times m-reduction [i] would yield (59, 85, 52658)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 258540 067960 993884 013239 645555 074401 153124 759704 316874 983625 > 585 [i]